$C_\lambda$- Extended oscillator algebra and $d$-orthogonal polynomials

TL;DR

This paper constructs an analytic realization of the $C_$-extended oscillator algebra using difference-differential operators and introduces new $d$-orthogonal polynomials extending Hermite and Laguerre polynomials, analyzing their properties and matrix realizations.

Contribution

It provides a systematic construction of $d$-orthogonal polynomials related to the $C_$-extended oscillator algebra and explores their algebraic and analytical properties.

Findings

Derived recurrence relations and difference-differential equations for the polynomials.

Established raising and lowering operators for the polynomial families.

Constructed matrix realizations of the algebra using these polynomials.

Abstract

In this paper we first construct an analytic realization of the $C_\lambda$-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of $d$-orthogonal polynomials which are extensions of the Hermite and Laguerre polynomials. The underlying algebraic framework allowed us a systematic derivation of their main properties such as recurrence relations, difference-differential equations, lowering and rising operators and generating functions. Finally, we use these polynomials to construct a realization of the $C_\lambda$-extended oscillator by block matrices.

Equations

$$[a_{-},a_{+}]=1+\sum _{i=1}^{\lambda -1}{\beta }_i{s}^i,s^{\lambda }=1,$$

$$[N,a_{-}]=-a_{-},[N,a_{+}]=a_{+},[N,s]=0,$$

$$a_{-}s={\epsilon }_{\lambda }s{a}_{-}{a}_{+}s={\epsilon }_{\lambda }^{-1}s{a}_{+},{\epsilon }_{\lambda }=e^{\frac{2i\pi }{\lambda }},$$

$$N^{\ast }=N,a_{+}^{\ast }=a_{-},s^{\ast }=s^{-1},$$

$$\overline{{\beta }_i}={\beta }_{\lambda -i},i=1,\dots ,\lambda .$$

$$[a_{-},a_{+}]$$

$$\{a_{\pm },R\}$$

$$a_{-}=\frac{1}{\sqrt{2}}(Y_{\nu }+x),a_{+}=\frac{1}{\sqrt{2}}(-Y_{\nu }+x),$$

$$Y_{\nu }:=\frac{d}{dx}+\frac{\nu }{x}(1-R),$$

$$(R\psi )(x)=\psi (-x)$$

$$H_{\nu }$$

$${\lambda }_n=n+\nu +\frac{1}{2},n=0,1,2,\dots$$

$${\psi }_n^{(\nu )}(x)={\gamma }_n{e}^{-x^2/2}{H}_n^{(\nu )}(x),$$

$${\gamma }_n={\left(2^{2n}\Gamma ([\frac{n}{2}]+1)\Gamma ([\frac{n+1}{2})]+\nu +\frac{1}{2})\right)}^{-\frac{1}{2}}$$

$$\{\begin{array}{l}{H}_{2n}^{(\nu )}(x)=(-1{)}^n{2}^{2n}n!L_n^{\nu -\frac{1}{2}}(x^2),\\ H_{2n+1}^{(\nu )}(x)=(-1{)}^n{2}^{2n+1}n!x{L}_n^{\nu +\frac{1}{2}}(x^2).\end{array}$$

$${\int }_{\mathbb{R}}{H}_n^{(\nu )}(x)H_m^{(\nu )}(x)|x{|}^{2\nu }{e}^{-x^2}dx=\frac{1}{{\gamma }_n^2}{\delta }_{nm},$$

$$C_{\lambda }=\{1,s,s^2,\dots ,s^{\lambda -1}\}$$

$${\Pi }_i=\frac{1}{\lambda }\sum _{j=0}^{\lambda -1}{\epsilon }_{\lambda }^{-ij}{s}^j,i=0,\dots ,\lambda -1,{\epsilon }_{\lambda }=e^{\frac{2i\pi }{\lambda }}$$

$$H=\frac{1}{2}\{a_{-},a_{+}\}.$$

$$[a_{-},a_{+}]=1+\sum _{j=0}^{\lambda -1}{\hat{\beta }}_j{\Pi }_j,$$

$$[N,a_{-}]=-a_{-},[N,a_{+}]=a_{+},a_{+}{\Pi }_i={\Pi }_{i+1}{a}_{+},$$

$$\sum _{i=1}^{\lambda }{\Pi }_i=1,{\Pi }_i{\Pi }_j={\delta }_{ij}{\Pi }_i,{\Pi }_i^{†}={\Pi }_i,$$

$${\hat{\beta }}_j=\sum _{i=0}^{\lambda -1}{\epsilon }_{\lambda }^{ij}{\beta }_i,j=1,\dots ,\lambda -1,$$

$$[a_{-}^{n\lambda },a_{+}]=n\lambda a_{-}^{n\lambda -1}\text{and}[a_{-},a_{+}^{n\lambda }]=n\lambda a_{+}^{n\lambda -1}.$$

$$[a^n,b]=\sum _{i=0}^{n-1}{a}^{n-1-i}[a,b]a^i.$$

$$N|k\lambda +i⟩=(k\lambda +i)|k\lambda +i⟩,{\Pi }_j|k\lambda +i⟩={\delta }_{ij}|k\lambda +i⟩,$$

$$a_{+}|k\lambda +i⟩=\sqrt{k\lambda +i+1+{\hat{\beta }}_{i+1}}|k\lambda +i+1⟩,$$

$$a_{-}|k\lambda +i⟩=\sqrt{k\lambda +i+{\hat{\beta }}_i}|k\lambda +i-1⟩.$$

$$\sum _{l=0}^{\lambda -1}{\nu }_l=0.$$

$$(Sf)(z):=f({\epsilon }_{\lambda }z).$$

$$Y_{\nu }:=\frac{d}{dz}+\frac{1}{z}\sum _{j=1}^{\lambda -1}{\nu }_j{S}^j.$$

$$S_{i}z=(z_1,\dots ,\epsilon z_i,\dots ,z_N).$$

$$Y_i=\frac{\partial }{\partial z_i}+{\kappa }_0\sum _{j\ne i}\sum _{j=0}^{r-1}\frac{1-S_i^{-j}(i,j)S_i^j}{z_i-{\epsilon }^j{z}_j}+\sum _{j=1}^{r-1}{\kappa }_j\sum _{l=0}^{r-1}\frac{{\epsilon }^{-rl}{s}_i^l}{z_i},$$

$${\hat{\nu }}_s=\sum _{l=0}^{\lambda -1}{\nu }_l{\epsilon }_{\lambda }^{sl}.$$

Full text

$\displaystyle C_{\lambda}$- Extended oscillator algebra and $\displaystyle d$-orthogonal polynomials

Fethi Bouzeffour ∗,⋄, Wissem Jedidi §,⋄

∗ Department of mathematics, College of Sciences, King Saud University, P. O Box 2455 Riyadh 11451, Saudi Arabia.

§ Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

⋄ Université de Tunis El Manar, Faculté des Sciences de Tunis, LR11ES11 Laboratoire d’Analyse Mathématiques et Applications, 2092, Tunis, Tunisia.

fbouzaffour@ksu.edu.sa; wissem.jedidi@fst.utm.tn

Abstract: In this paper we first construct an analytic realization of the $\displaystyle C_{\lambda}$-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of $\displaystyle d$-orthogonal polynomials which are extensions of the Hermite and Laguerre polynomials. The underlying algebraic framework allowed us a systematic derivation of their main properties such as recurrence relations, difference-differential equations, lowering and rising operators and generating functions. Finally, we use these polynomials to construct a realization of the $\displaystyle C_{\lambda}$-extended oscillator by block matrices. Keywords: vector orthogonal polynomials, deformed oscillator algebra .

1. Introduction

In [20], Quesne defined the $\displaystyle C_{\lambda}$-extended oscillator algebra ($\displaystyle\lambda$ being an integer bigger than $\displaystyle 1$) as the associative algebra generated by four elements $\displaystyle a_{-},$ $\displaystyle a_{+},$ $\displaystyle N$, $\displaystyle s$ that obey the commutation relations

[TABLE]

where the constants structure $\displaystyle\beta_{1},\dots,\,\beta_{\lambda-1}$ are complex numbers restricted by the conditions

[TABLE]

The connection between orthogonal polynomials and harmonic oscillator as well as the quantum algebra is well known [28, 21]. The oscillator algebra of creation, annihilation, and number operators plays a central role in the investigation of many class of orthogonal polynomials, and provides a useful tool to derive their operational properties such as recurrence relations, difference equation, lowering and rising operators and generating functions. It particular, when $\displaystyle\lambda=2$, the $\displaystyle C_{\lambda}$-extended Heisenberg algebra is reduced to the Calogero–Vasiliev algebra [16], which is given by the generators $\displaystyle a_{-},\,a_{+},\,R$, $\displaystyle 1$ that satisfy the commutation relations

[TABLE]

A realization of the Calogero–Vasiliev algebra working in the Schrödinger representation, $\displaystyle\Psi=\Psi(x)$, is given by the operators

[TABLE]

where $\displaystyle Y_{\nu}$ is the Dunkl operator (corresponding to the root system $\displaystyle A_{1}$, see [13, Definition 4.4.2)]), which is a differential-difference operator, depending on a parameter $\displaystyle\nu\in\mathbb{R}$ and acting on $\displaystyle C^{\infty}(\mathbb{R})$ as follows:

[TABLE]

where $\displaystyle R$ is the Klein operator acting on function $\displaystyle\psi$ of the real variable $\displaystyle x$ as

[TABLE]

(see [16] for more details). Note that the operator $\displaystyle Y_{\nu}$ is also related by a simple similarity transformation to the Yang-Dunkl operator used in [16], were the authors show that the Calogero–Vasiliev algebra is intimately related to parabosons and parafermions, and to the $\displaystyle osp(1|2)$ and $\displaystyle osp(2|2)$ superalgebras.

The Hamiltonian takes the form

[TABLE]

and the wave functions corresponding to the well-known eigenvalues

[TABLE]

are given by

[TABLE]

where

[TABLE]

and $\displaystyle H^{(\nu)}_{n}(x)$ is the generalized Hermite polynomial introduced by Szegö [26] and obtained in [26] from Laguerre polynomials $\displaystyle L_{n}^{\nu}(x)$ by the means:

[TABLE]

It is well known that for $\displaystyle\nu>-\frac{1}{2},$ these polynomials satisfy the orthogonality relations:

[TABLE]

where $\displaystyle\gamma_{n}$ is given in (11). Many of the known generalized Hermite polynomials are also the eigenfunctions of the energy operator for a deformed oscillator (see [15, 3]).

In this paper, we discuss the connection of some class of $\displaystyle d$-orthogonal polynomials with the $\displaystyle C_{\lambda}$-extended oscillator algebra (for $\displaystyle\lambda=d+1$). The $\displaystyle d$-orthogonal polynomials can be obtained from general multiple orthogonal polynomials under some restrictions upon their parameters (see [27]). Applications of $\displaystyle d$-orthogonal polynomials include the simultaneous Padé approximation problem where the multiple orthogonal polynomials appear (see [6]). Also they play important role in random matrix theory (see [4]). The $\displaystyle d$-orthogonal polynomials have been intensively studied in the last 30 years due to their intriguing properties and applications (see [5, 7] and further references in the literature).

One problem that deserves attention is to relate $\displaystyle d$-orthogonal polynomials to some oscillator algebras. In [29, 8, 9], the authors have found some $\displaystyle d$-orthogonal polynomials related to the deformed harmonic oscillator and share a number of properties with the classical orthogonal polynomials. In this paper, we discuss the connection of some class of $\displaystyle d$-orthogonal polynomials with the $\displaystyle C_{\lambda}$-extended oscillator algebra $\displaystyle(\lambda=d+1)$. We use a Klein type operator $\displaystyle S$ of finite order to construct a realization of the $\displaystyle C_{\lambda}$-extended oscillator algebra in terms of difference-differential operator and a system of vector orthogonal polynomials obtained from $\displaystyle d$-orthogonal polynomials, in order to provide realizations of the $\displaystyle C_{\lambda}$-extended oscillator by block matrices. Note that the obtained $\displaystyle d$-orthogonal polynomials are eigenfunctions of the operator $\displaystyle S$.

The paper is organized as follows. In Section 2, we review the definition of the $\displaystyle C_{\lambda}$-extended oscillator algebra and introduce a Bergmann realization associated to this oscillator. In section 3, we deal with the formalism of the exponential of the Dunkl type operators, which are the most commonly used operators in the context of evolution problems. We establish the relevant rules to the action of an exponential operator on a given function and those for the disentanglement of exponential operators. We also establish there the most important properties of $\displaystyle d$-orthogonal polynomials. In section 4, we construct a family of vector orthogonal polynomial and analyse the effect of the generators of the $\displaystyle C_{\lambda}$-extended algebra on it.

2. Bergmann realization of the $\displaystyle C_{\lambda}$-extended oscillator algebra

2.1. The $\displaystyle C_{\lambda}$-extended oscillator algebra

The element $\displaystyle s$ introduced in (1), is a generator of the cyclic group

[TABLE]

of order $\displaystyle\lambda$. The primitive idempotents related to this group are denoted by $\displaystyle\Pi_{0},\,\Pi_{1},\dots,\,\Pi_{\lambda-1}$ and are given by

[TABLE]

and the bosonic $\displaystyle C_{\lambda}$-extended oscillator Hamiltonian is defined by

[TABLE]

According to [18], it may therefore be rewritten in terms of $\displaystyle a_{-},$ $\displaystyle a_{+},$ $\displaystyle N$, $\displaystyle\Pi_{0},\dots,\,\Pi_{\lambda-1}$ as follows:

[TABLE]

where, according to condition (5), the discrete Fourier transforms $\displaystyle\widehat{\beta}_{0},\dots\widehat{\beta}_{\lambda-1},$ are real numbers given by

[TABLE]

and are restricted by the condition $\displaystyle\sum_{i=0}^{\lambda-1}\widehat{\beta}_{i}=0$.

Next proposition will be needed in the sequel and gives a relation in the $\displaystyle C_{\lambda}$-extended oscillator algebra.

Proposition 1**.**

*Let $\displaystyle n\in\mathbb{N},$ we have

1. \displaystyle[a_{-}^{n},a_{+}]=\big{(}n+\sum_{i=0}^{\lambda-1}\beta_{i}\frac{\varepsilon_{\lambda}^{ni}-1}{\varepsilon_{\lambda}^{i}-1}s^{i}\big{)}\,a_{-}^{n-1},

2. \displaystyle[a_{-},a_{+}^{n}]=a_{+}^{n-1}\,\big{(}n+\sum_{i=0}^{\lambda-1}\beta_{i}\frac{\varepsilon_{\lambda}^{ni}-1}{\varepsilon_{\lambda}^{i}-1}s^{i}\big{)},

3. $\displaystyle[N,a_{-}^{n}]=-na_{-}^{n-1},\qquad[N,a_{+}^{n}]=na_{+}^{n-1}.$

In particular, it holds that

[TABLE]

Proof.

The proof is easy and follows from the identity

[TABLE]

∎

According to [18], the $\displaystyle C_{\lambda}$-extended oscillator algebra possesses a canonical irreducible representation defined on the orthonormal basis $\displaystyle|n\rangle,\,\,n=0\,1,\,2,\,.\dots$, endowed with the following actions:

[TABLE]

2.2. Dunkl type operator

Let $\displaystyle{\mathrm{\nu}}=(\nu_{1},\dots,\nu_{\lambda-1})$, where $\displaystyle\nu_{1},\dots,\nu_{\lambda-1}$ are complex numbers satisfying the condition

[TABLE]

Consider the Klein-type reflection $\displaystyle S$ acting on functions $\displaystyle f(z)$ of complex variable $\displaystyle z$ as follows

[TABLE]

and its associated differential-difference operator

[TABLE]

Recall that the complex reflection group $\displaystyle G(r,1,N)$ consists of the $\displaystyle N\times N$ permutation matrices with the nonzero entries being powers of $\displaystyle\varepsilon=e^{\frac{2i\pi}{r}}.$ The group $\displaystyle G(r,1,N)$ is also generated by the transpositions $\displaystyle(i,i+1),\,\,i=1,\,\dots,\,N-1,$ and by the complex reflections $\displaystyle S_{i}$ defined by

[TABLE]

When $\displaystyle N=1$, it happens that:

• •

the group $\displaystyle G(r,1,N)$ is isomorphic to the cyclic group $\displaystyle C_{\lambda}$ defined in (14);

• •

the operator defined in (25) is a particular case of the complex Dunkl operator $\displaystyle Y_{i}$ associated to $\displaystyle G(r,1,N)$ given by [13]:

[TABLE]

where $\displaystyle\kappa_{0},\,\kappa_{1},\dots,\kappa_{r}$ are real numbers.

2.3. Some results on deformed factorial numbers

Let $\displaystyle\nu_{i},\,i=0,\,1,\cdots,\,\lambda-1$ be a sequence of complex numbers and $\displaystyle\widehat{\nu}_{s}$ their discrete Fourier transforms defined as

[TABLE]

One can notice that

[TABLE]

Then, for nonnegative integers $\displaystyle n$, we introduce the deformed numbers $\displaystyle[n]_{\mathbf{\nu}}$ and the deformed factorial numbers by

[TABLE]

For

[TABLE]

and for $\displaystyle 0\leq s\leq\lambda-1$, we need to introduce the multi-index numbers

[TABLE]

We obtain the following result:

Proposition 2**.**

The initial problem

[TABLE]

has a unique analytic solution given by

[TABLE]

Furthermore, the generalized exponential function $\displaystyle\mathcal{E}_{\lambda}(\varrho\,.\,,\nu)$ has the following hypergeometric representation

[TABLE]

Proof.

It is easily seen from the action of the operator $\displaystyle Y_{\nu}$ on monomials $\displaystyle z^{n}$:

[TABLE]

that the function $\displaystyle\mathcal{E}_{\lambda}(\varrho\,.\,,\nu)$ is the unique solution of the system (34). To prove (35), it suffice to write $\displaystyle[n\lambda+s]_{\nu}!\,$ in terms of the Pochhammer Symbol $\displaystyle(a)_{n}$ defined for $\displaystyle a\in\mathbb{C}$ and $\displaystyle n=1,\,\dots\,$ by

[TABLE]

For $\displaystyle s=1,\,2,\,\dots\lambda-1,$ and with $\displaystyle\alpha_{k}$ defined in (31), we get

[TABLE]

Similarly,

[TABLE]

If we decompose the sum (34) in the form

[TABLE]

we obtain

[TABLE]

and this allows to conclude. ∎

2.4. Bergmann realization

Let $\displaystyle G^{m,n}_{p,q}$ be the Meijer’s $\displaystyle G$-function [22], $\displaystyle m_{s}$ be the measure defined by

[TABLE]

and $\displaystyle L_{\nu,\lambda}^{2}(\mathbb{C})$ be the space of measurable functions $\displaystyle f$ on $\displaystyle\mathbb{C}$ that satisfy

[TABLE]

The generalized Bergmann space $\displaystyle\mathfrak{B}_{\nu,\lambda}(\mathbb{C})$ is the pre-Hilbert space of analytic functions in $\displaystyle L_{\nu,\lambda}^{2}(\mathbb{C})$, equipped with the inner product

[TABLE]

where

[TABLE]

Theorem 1**.**

If $\displaystyle f,g\in\mathfrak{B}_{\nu,\lambda}(\mathbb{C})$ expanded in the form

[TABLE]

then, their inner product is given by

[TABLE]

Proof.

Let $\displaystyle 0\leq s,s^{\prime}\leq\lambda-1$ and $\displaystyle n,\,m=0,\,\dots.$ From (43), it is clear that for $\displaystyle s\neq s^{\prime}$, we have

[TABLE]

hence,

[TABLE]

For $\displaystyle s=s^{\prime}$, we have

[TABLE]

Now using the following well known formula [22]

[TABLE]

we obtain

[TABLE]

∎

It is oblivious that the space $\displaystyle\mathfrak{B}_{\nu,\lambda}(\mathbb{C})$ is a Hilbert space equipped with the inner product (41) and the monomials

[TABLE]

constitute an orthonormal basis for this space. The space $\displaystyle\mathfrak{B}_{\nu,\lambda}(\mathbb{C})$ has the kernel function

[TABLE]

Proposition 3**.**

$\displaystyle(i)$* Let $\displaystyle Z$ be the multiplication operator ($\displaystyle(Zf)(z):=zf(z)$. The operators $\displaystyle Y_{\nu}$ and $\displaystyle Z$ are closed densely defined operators on the space $\displaystyle\mathfrak{B}_{\nu,\lambda}(\mathbb{C})$ on the common domain*

[TABLE]

Furthermore, for $\displaystyle f,g\in D$ one has

[TABLE]

Proposition 4**.**

*The following holds:

$\displaystyle(i)$*

[TABLE]

$\displaystyle(ii)$**

[TABLE]

Now, assume that the complex numbers $\displaystyle\nu_{i}$ are restricted by the conditions

[TABLE]

The case where the structure constants $\displaystyle\beta_{i}$ in (1) take the values

[TABLE]

we get a convenient one variable model of the representation of $\displaystyle C_{\lambda}$-extended oscillator algebra given by the Hilbert space $\displaystyle\mathfrak{B}_{\nu,\lambda}(\mathbb{C})$ and the action of the $\displaystyle C_{\lambda}$-extended oscillator algebra is given by

[TABLE]

and the related bosonic extended Hamiltonian takes the form

[TABLE]

3. Generalized Hermite polynomials

For fixed integer $\displaystyle m$, Gould and Hopper [14] defined the generalized Hermite polynomials $\displaystyle H_{n}(x,m)$ by the operational identity

[TABLE]

Similarly, a new family of generalized Hermite polynomials $\displaystyle\{H^{(\lambda,\nu)}_{n}(x)\}_{n=0}^{\infty}$ can be determined by means of the following operational formula

[TABLE]

Due to the fact that $\displaystyle Y_{\nu}$ reduces the degrees of the polynomials , see equation (36) then the latter series (54) contains only a finite number of terms. The operational definition (54) greatly simplifies the study of the generalized Hermite polynomials. From Proposition 4, we deduce that

[TABLE]

and that

[TABLE]

Proposition 1**.**

The following holds true

[TABLE]

Proof.

Performing the Dunkl operator $\displaystyle Y_{\nu}$ of both sides of (54) with respect to $\displaystyle x$, we obtain

[TABLE]

The equation (56) can be exploited to derive the operational formula

[TABLE]

Hence

[TABLE]

∎

We also obtain

Theorem 2**.**

The polynomials $\displaystyle H^{(\lambda,\nu)}_{n}(x)$ satisfy the following higher order differential-difference equations

[TABLE]

Proof.

Applying the operator $\displaystyle Y_{\nu}$ to the both side of (58), we get the following differential-difference equation:

[TABLE]

∎

Proposition 2**.**

The polynomials $\displaystyle H^{(\lambda,\nu)}_{n}$ are generated by the series

[TABLE]

Furthermore,

[TABLE]

Proof.

According to (58) and (33), we can write

[TABLE]

Hence, the polynomials $\displaystyle\{H^{(\lambda,\nu)}_{n}(x)\}$ are generated by

[TABLE]

An explicit formula for the generalized Hermite polynomials $\displaystyle H^{(\lambda,\nu)}_{n}(x)$ is obtained by expanding in power series the generation function given in (61):

[TABLE]

Substituting $\displaystyle n=k\lambda+m$, then $\displaystyle 0\leq k\leq[n/\lambda]$ and we

[TABLE]

Finally,

[TABLE]

∎

Proposition 3**.**

The polynomials $\displaystyle H^{(\lambda,\nu)}_{n}$ satisfy the following three terms recurrence relations

[TABLE]

Proof.

By taking into account (57) and (58) we get the three terms recurrence relations

[TABLE]

∎

4. Matrix realization of the $\displaystyle C_{\lambda}$-extended oscillator

In sequel we assume that $\displaystyle d=\lambda-1$. We propose here a realization of the $\displaystyle C_{\lambda}$-extended oscillator by matrices.

Let us first clarify the relationship between the notion of d-orthogonality of a family of polynomials and orthogonality of vector polynomials. Recall that $\displaystyle d$-orthogonal polynomials are system $\displaystyle\{P_{n}(x)\}$ of monic polynomials (with $\displaystyle\deg P_{n}=n$) such that there exists a vector linear functional $\displaystyle\,\mathcal{U}=\begin{bmatrix}u_{0}\dots u_{d-1}\end{bmatrix}^{T}$ satisfying the conditions:

[TABLE]

where $\displaystyle\langle u,P\rangle$ is the effect of a linear functional $\displaystyle u$ on a polynomial $\displaystyle P$ and $\displaystyle[x]$ denotes the greatest integer function. Note that the case $\displaystyle d=1$ corresponds to the ordinary notion of orthogonal polynomials. According to [24, 25] the vector orthogonality relations is equivalent to the existence of a linear $\displaystyle(d+2)$-term recurrence relation

[TABLE]

with constants $\displaystyle a_{j}(n)$, $\displaystyle a_{j}(d)\neq 0$.

• •

A system of polynomials $\displaystyle\{P_{n}\}$ is said to be $\displaystyle d$–symmetric when it verifies

[TABLE]

• •

We say that the vector of linear functionals $\displaystyle\,\mathcal{U}=\begin{bmatrix}u_{1},\,\dots,\,u_{d}\end{bmatrix}^{T}$ is said to be $\displaystyle d$–symmetric when the moments of its entries satisfy, for every $\displaystyle n\geq 0$,

[TABLE]

According to [11], for every sequence of monic polynomials $\displaystyle\{P_{n}\}$, $\displaystyle d$–orthogonal with respect to the *vector of linear functionals * $\displaystyle\mathcal{U}=\begin{bmatrix}u_{0}\dots u_{d-1}\end{bmatrix}^{T}$, the following statements are equivalent :

• $\displaystyle(i)$

the vector of linear functionals $\displaystyle\mathcal{U}$ is $\displaystyle d$–symmetric;

• $\displaystyle(ii)$

the sequence $\displaystyle\{P_{n}\}$ is $\displaystyle d$–symmetric;

• $\displaystyle(iii)$

the sequence $\displaystyle\{P_{n}\}$ satisfies

[TABLE]

with $\displaystyle P_{n}(x)=x^{n}$ for $\displaystyle 0\leq n\leq d$.

Within this context, we obtain the following result on the generalized Hermite polynomials:

Theorem 3**.**

For $\displaystyle d=\lambda-1$, the family $\displaystyle\{H^{(\lambda,\nu)}_{n}(x)\}$ are $\displaystyle d$-orthogonal polynomials with respect to the functionals $\displaystyle u_{0},\,\dots,\,u_{d-1}$, which are determined by their moments:

[TABLE]

where

[TABLE]

Proof.

From (64), the polynomials $\displaystyle\{H^{(\lambda,\nu)}_{n}(x)\}$ satisfy a $\displaystyle(\lambda+1)$-term recursion relation of the form (70) and then, are $\displaystyle d$-orthogonal $\displaystyle d$-symmetric polynomials. Furthermore there exist $\displaystyle\lambda-1$ symmetric functionals $\displaystyle u_{k},k=0,\ldots,d-1$, on the space of all polynomials $\displaystyle\mathcal{P}$ such that

[TABLE]

To determine the moments of the functional $\displaystyle u_{k}$, we will need the following inversion formula

[TABLE]

To prove (74), we can use the generating function (61) to first obtain

[TABLE]

then, substituting $\displaystyle m=n\lambda+k$, then $\displaystyle 0\leq n\leq[m/\lambda]$, we get

[TABLE]

and it remains to compare the coefficient of $\displaystyle t^{m}$ in the two expansion (34) and (76) of $\displaystyle\mathcal{E}_{\lambda}(xt,\nu)$.

Following Maroni [24], the moments of the functionals $\displaystyle u_{k},k=0,\ldots,d-1$ related to $\displaystyle\{H^{(\lambda,\nu)}_{n}(x)\}$ can be obtained from the inversion formula (74)

[TABLE]

It remains to prove the integral representation of the functionals $\displaystyle u_{k}$, so it suffices to substitute

[TABLE]

in the integral (47), in order to get

[TABLE]

The latter yields

[TABLE]

where

[TABLE]

so that

[TABLE]

∎

Let $\displaystyle\{P_{n}\}$ be a $\displaystyle d$-symmetric family of $\displaystyle d$-orthogonal polynomials with respect to the vector of functionals \displaystyle\,\mathcal{U}=\big{[}u_{0},\dots,u_{d-1}\big{]}^{T} . We define the family of vector polynomials $\displaystyle\{\mathbb{P}_{n}\}$ related to $\displaystyle\{P_{n}\}$ as follows :

[TABLE]

and we extend the action of the vector of functionals $\displaystyle\,\mathcal{U}$ in vector polynomials $\displaystyle\{\mathbb{P}_{n}\}$ as follows:

[TABLE]

Within this context, $\displaystyle\{\mathbb{P}_{n}\}$ is said to be a vector orthogonal polynomial sequence with respect to the vector of functionals $\displaystyle\mathcal{U}$, if

[TABLE]

where $\displaystyle\Delta_{n}$ is a regular upper triangular $\displaystyle d\times d$ matrix.

We can see easily that the $\displaystyle d$-orthogonality of a family of polynomials $\displaystyle\{P_{n}\}$ defined in (65) is equivalent to the vector orthgonality of the related family $\displaystyle\{\mathbb{P}_{n}\}$ defined as above. For a deeper account of the theory (in a more general framework, considering quasi–diagonal multi–indices) we refer the reader to [1].

Let consider the family of vector polynomials $\displaystyle\{\mathbb{H}^{(\lambda,\nu)}_{n}(x)\}$ given by

[TABLE]

where

[TABLE]

From (64), we easily check the three terms recurrence relations

[TABLE]

where

[TABLE]

We also extend the action of the operator $\displaystyle Y_{\nu}$ on the vector polynomial $\displaystyle\mathbb{H}^{(\lambda,\nu)}_{n}(x)$ through

[TABLE]

Theorem 4**.**

Under the above notations, we have

[TABLE]

with $\displaystyle\mathbb{H}^{(\lambda,\nu)}_{-1}=\begin{bmatrix}0,&\cdots,&0\end{bmatrix}^{T}$, $\displaystyle\mathbb{H}^{(\lambda,\nu)}_{0}=\begin{bmatrix}\widetilde{H}^{(\lambda,\nu)}_{0}(x),&\cdots,&\widetilde{H}^{(\lambda,\nu)}_{d-1}(x)\end{bmatrix}^{T}$, and matrix coefficients $\displaystyle A_{n}$, $\displaystyle B_{n}$, $\displaystyle C_{n},$ $\displaystyle R_{n}\in\mathcal{M}_{d\times d}$ given by

[TABLE]

The position operator $\displaystyle x$ can be identified with the following Bloc matrix

[TABLE]

and the momentum operator $\displaystyle Y_{\nu}$ with

[TABLE]